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R. Ahrens et al.
2.3.2 The Energy and Concentration Equations
As a next, natural step, the Eq. (1.2) for the energy balance and (1.3) for the concentration balance can be solved. Again, due to the numerical complexity, a monolithic
solution of this system of equations does not seem to be attractive. Instead, the equations are solved individually, by using the currently available data, e.g.,
1. ∂ t T n − D T ΔT n + u n · ∇T n = F ener,growth (c n−1 , T n−1 , f n−1 ) ,
2.
∂ t c n − D c Δc n + u n · ∇c n = F conc,growth (c n−1 , T n , f n−1 ) ,
where still the temporal derivatives have to be discretized. In this approach, one
has to solve two linear equations. The individual solution of these equations can be
iterated by using in the second iteration the temperature and concentration solution
computed in the first iteration and so on.
In many applications, in particular in crystallization processes, the diffusion
parameters in (1.2) and (1.3) are smaller by several orders of magnitude compared
with the size of the velocity field. This situation is called convection-dominated and
there is a similar difficulty as for turbulent flows: there are important features of
the solution, so-called layers, that cannot be resolved on affordable grids. As for
turbulent flows, standard numerical discretizations fail in this situation and the use
of a so-called stabilized discretization is necessary, e.g., see [55]. There are many
proposals for stabilized discretizations in the literature. In the context of the coupled
system (1.2) and (1.3), it is essential that the numerical solution computed with the
stabilized method must not possess unphysical values, so-called spurious oscillations, or it is allowed to exhibit only negligible spurious oscillations. This property
is important because the computed solutions serve as data in other equations, for
certain coefficients, and if the numerical solutions have spurious oscillations, then
non-physical coefficients in other equations might be computed. At any rate, it was
noted in [36] for a precipitation process that using a stabilized discretization that
does not sufficiently suppress spurious oscillations usually leads to a blow-up of the
simulations of the coupled system in finite time.
As a matter of fact, many of the proposed stabilized schemes lead to numerical
solutions with non-negligible spurious oscillations, e.g., see the numerical assessment in [39]. Some schemes that satisfy the requirement with respect to the spurious
oscillations are the followings:
• finite difference methods
– upwind; very diffusive and very inaccurate,
– FCT (flux-corrected transport) schemes [14];
– ENO (essentially non-oscillatory) [32], WENO (weighted ENO) [45]; much
more accurate, small spurious oscillations possible,
• finite element methods
– linear FEM-FCT [42]; often good compromise between accuracy and efficiency,
– FEM-FCT [43, 46]; nonlinear method, often quite accurate,
R. Ahrens et al.
2.3.2 The Energy and Concentration Equations
As a next, natural step, the Eq. (1.2) for the energy balance and (1.3) for the concentration balance can be solved. Again, due to the numerical complexity, a monolithic
solution of this system of equations does not seem to be attractive. Instead, the equations are solved individually, by using the currently available data, e.g.,
1. ∂ t T n − D T ΔT n + u n · ∇T n = F ener,growth (c n−1 , T n−1 , f n−1 ) ,
2.
∂ t c n − D c Δc n + u n · ∇c n = F conc,growth (c n−1 , T n , f n−1 ) ,
where still the temporal derivatives have to be discretized. In this approach, one
has to solve two linear equations. The individual solution of these equations can be
iterated by using in the second iteration the temperature and concentration solution
computed in the first iteration and so on.
In many applications, in particular in crystallization processes, the diffusion
parameters in (1.2) and (1.3) are smaller by several orders of magnitude compared
with the size of the velocity field. This situation is called convection-dominated and
there is a similar difficulty as for turbulent flows: there are important features of
the solution, so-called layers, that cannot be resolved on affordable grids. As for
turbulent flows, standard numerical discretizations fail in this situation and the use
of a so-called stabilized discretization is necessary, e.g., see [55]. There are many
proposals for stabilized discretizations in the literature. In the context of the coupled
system (1.2) and (1.3), it is essential that the numerical solution computed with the
stabilized method must not possess unphysical values, so-called spurious oscillations, or it is allowed to exhibit only negligible spurious oscillations. This property
is important because the computed solutions serve as data in other equations, for
certain coefficients, and if the numerical solutions have spurious oscillations, then
non-physical coefficients in other equations might be computed. At any rate, it was
noted in [36] for a precipitation process that using a stabilized discretization that
does not sufficiently suppress spurious oscillations usually leads to a blow-up of the
simulations of the coupled system in finite time.
As a matter of fact, many of the proposed stabilized schemes lead to numerical
solutions with non-negligible spurious oscillations, e.g., see the numerical assessment in [39]. Some schemes that satisfy the requirement with respect to the spurious
oscillations are the followings:
• finite difference methods
– upwind; very diffusive and very inaccurate,
– FCT (flux-corrected transport) schemes [14];
– ENO (essentially non-oscillatory) [32], WENO (weighted ENO) [45]; much
more accurate, small spurious oscillations possible,
• finite element methods
– linear FEM-FCT [42]; often good compromise between accuracy and efficiency,
– FEM-FCT [43, 46]; nonlinear method, often quite accurate,